holt algebra 2 5-5 complex numbers and roots define and use imaginary and complex numbers. solve...
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Holt Algebra 2
5-5 Complex Numbers and Roots
Define and use imaginary and complex numbers.
Solve quadratic equations with complex roots.
Objectives
Holt Algebra 2
5-5 Complex Numbers and Roots
You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions.
However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as . You can use the imaginary unit to write the square root of any negative number.
Holt Algebra 2
5-5 Complex Numbers and Roots
Holt Algebra 2
5-5 Complex Numbers and Roots
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.
Holt Algebra 2
5-5 Complex Numbers and Roots
Express the number in terms of i.
Holt Algebra 2
5-5 Complex Numbers and Roots
Solve the equation.
Take square roots.
Express in terms of i.
Checkx2 = –144
–144–144–144
(12i)2
144i 2
144(–1)
x2 = –144
–144–144–144144(–1)
144i 2
(–12i)2
Holt Algebra 2
5-5 Complex Numbers and Roots
Solve the equation.
Check 5x2 + 90 = 0
000
5(18)i 2 +9090(–1) +90
5x2 + 90 = 0
Holt Algebra 2
5-5 Complex Numbers and Roots
x2 = –36
Solve and check the equations.
x2 + 48 = 0 9x2 + 25 = 0
Holt Algebra 2
5-5 Complex Numbers and Roots
Every complex number has a real part a and an imaginary part b.
A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = . The set of real numbers is a subset of the set of complex numbers C.
Holt Algebra 2
5-5 Complex Numbers and Roots
Real numbers are complex numbers where b = 0. Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pure imaginary numbers.
Two complex numbers are equal if and only if their real
parts are equal and their imaginary parts are equal.
Holt Algebra 2
5-5 Complex Numbers and Roots
Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true .
Equate the real parts.
4x + 10i = 2 – (4y)i
Real parts
Imaginary parts
4x = 2
Solve for y.Solve for x.
Equate the imaginary parts.
10 = –4y
Holt Algebra 2
5-5 Complex Numbers and Roots
Find the values of x and y that make each equation true.
2x – 6i = –8 + (20y)i –8 + (6y)i = 5x – i 6
Holt Algebra 2
5-5 Complex Numbers and Roots
Find the zeros of the function.
x2 + 10x + = –26 +
f(x) = x2 + 10x + 26
Add to both sides.
x2 + 10x + 26 = 0 Set equal to 0.
Rewrite.
x2 + 10x + 25 = –26 + 25
(x + 5)2 = –1 Factor.
Take square roots.
Simplify.
Think “Complete the Square”
Holt Algebra 2
5-5 Complex Numbers and Roots
g(x) = x2 + 4x + 12
Find the zeros of the function. Think “Complete the Square”
h(x) = x2 – 8x + 18
Holt Algebra 2
5-5 Complex Numbers and Roots
The solutions and are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi.
Find each complex conjugate.
A. 8 + 5i B. 6i
0 + 6i8 + 5i
8 – 5i
Write as a + bi.
Find a – bi.
Write as a + bi.
Find a – bi.
Simplify.
0 – 6i
–6i
Holt Algebra 2
5-5 Complex Numbers and Roots
Find each complex conjugate.
A. 9 – i
9 + (–i)
9 – (–i)
Check It Out! Example 5
B.
C. –8i0 + (–8)i
0 – (–8)i
8i
9 + i
Holt Algebra 2
5-5 Complex Numbers and Roots
Lesson Quiz
Solve each equation.
2. 3x2 + 96 = 0 3. x2 + 8x +20 = 0
1. Express in terms of i.
4. Find the values of x and y that make the equation 3x +8i = 12 – (12y)i true.
5. Find the complex conjugate of